USAAAO — 2023 First Round

A comet passes near the Sun on a parabolic orbit. While it's passing near the Sun with orbital velocity V , the Sun's heat causes the comet to melt, and it shatters into many small fragments. The fragments move away uniformly in all directions (in the comet's reference frame) with velocity v ≪ V . What fraction of the fragments will escape the solar system? Ignore any forces other than the Sun's gravity.